Majorant estimates for partial sums of multiple fourier series of functions from Orlicz spaces vanishing on some set

1999 ◽  
Vol 65 (6) ◽  
pp. 694-700
Author(s):  
O. K. Ivanova
Keyword(s):  
Fourier Series ◽  
Orlicz Spaces ◽  
Multiple Fourier Series ◽  
Partial Sums ◽  
Series Of Functions

STRUCTURAL AND GEOMETRIC CHARACTERISTICS OF SETS OF CONVERGENCE AND DIVERGENCE OF MULTIPLE FOURIER SERIES OF FUNCTIONS WHICH EQUAL ZERO ON SOME SET

2004 ◽  
Vol 02 (02) ◽  
pp. 187-195 ◽  
Author(s):  
I. L. BLOSHANSKII
Keyword(s):  
Fourier Series ◽  
Positive Measure ◽  
Linear Subspace ◽  
Multiple Fourier Series ◽  
Partial Sums ◽  
Geometric Characteristics ◽  
Convergence Almost Everywhere ◽  
Almost Everywhere ◽  
Dimensional Cube ◽  
Series Of Functions

Let E be an arbitrary set of positive measure in the N-dimensional cube TN=(-π,π)N⊂ℝN, N≥1, and let f(x)=0 on E. Let [Formula: see text] be some linear subspace of L1(TN). We investigate the behavior of rectangular partial sums of multiple trigonometric Fourier series of a function f on the sets E and TN\E depending on smoothness of the function f (i.e. of the space [Formula: see text]), and, as well, of structural and geometric characteristics of the set E (SGC(E)). Thus, we are describing pairs [Formula: see text]. It is convenient to formulate and investigate the posed question in terms of generalized localization almost everywhere (GL) and weak generalized localization almost everywhere (WGL). This means that for the multiple Fourier series of a function f, that equals zero on the set E, convergence almost everywhere is investigated on the set E (GL), or on some of its subsets E1⊂E, of positive measure (WGL).

scholarly journals Generalized localization for spherical partial sums of multiple Fourier series

2019 ◽  
Vol 489 (1) ◽  
pp. 7-10
Author(s):  
R. R. Ashurov
Keyword(s):  
Fourier Series ◽  
Multiple Fourier Series ◽  
Partial Sums ◽  
Localization Principle ◽  
Open Set ◽  
Almost Everywhere

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2-class is proved, that is, if f L2 (ТN) and f = 0 on an open set ТN then it is shown that the spherical partial sums of this function converge to zero almost - ​everywhere on . It has been previously known that the generalized localization is not valid in Lp (TN) when 1 p 2. Thus the problem of generalized localization for the spherical partial sums is completely solved in Lp (TN), p 1: if p 2 then we have the generalized localization and if p 2, then the generalized localization fails.

scholarly journals On the Localization of the Riesz Means of Multiple Fourier Series of Distributions

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
A. A. Rakhimov
Keyword(s):  
Fourier Series ◽  
Compact Support ◽  
Sufficient Conditions ◽  
Multiple Fourier Series ◽  
Partial Sums ◽  
Riesz Means ◽  
Fourier Expansions

We study special partial sums of multiple Fourier series of distributions. We obtain sufficient conditions of summation of Riesz means of Fourier expansions of distributions with compact support.

Absolute convergence of multiple Fourier series of a function of p(n)-Λ-BV

2018 ◽  
Vol 25 (3) ◽  
pp. 481-491
Author(s):  
Rajendra G. Vyas
Keyword(s):  
Fourier Series ◽  
Absolute Convergence ◽  
Multiple Fourier Series ◽  
Series Of Functions ◽  
Sufficiency Conditions

AbstractIn this paper, we obtain sufficiency conditions for generalized β-absolute convergence ({0<\beta\leq 2}) of single and multiple Fourier series of functions of the class {\Lambda\text{-}\mathrm{BV}(p(n)\uparrow\infty,\varphi,[-\pi,\pi])} and the class {(\Lambda^{1},\Lambda^{2},\dots,\Lambda^{N})\text{-}\mathrm{BV}(p(n)\uparrow% \infty,\varphi,[-\pi,\pi]^{N})}, respectively.

Sign in / Sign up

Export Citation Format

Share Document